TODO:
knitr::opts_chunk$set(echo = TRUE,
cache = TRUE,
warning = FALSE,
message = FALSE)
set.seed(2430024)
date <- lubridate::now()
format(date, "%a, %d. %B %Y")
## [1] "Wed, 12. February 2020"
# Load Packages
library(broman)
library(tidyverse)
library(knitr)
library(googlesheets4)
library(scales)
library(ggpubr)
library(magick)
library(gridExtra)
library(bayesplot)
# google sheets setup
googlesheet_id <- "1-0mfTM47nF2etqPqvcQzp9xspgTLNHjp-cZ0E1qGAjs"
sheets_auth(email = "jenshuesers@gmail.com")
# Import Local Package "PEDIS"
unloadNamespace("pedis")
devtools::load_all("~/r-projects/proj_wunde/pedis/")
# Data Import
pedis <- pedis::read_all_pedis("~/r-projects/proj_wunde/pedis-diabetic-care/datasets")
desc_all <- descriptive_table_all(pedis)
sheets_write(data = desc_all,
ss = googlesheet_id,
sheet = "descr-table-all")
desc_any <- descriptive_table_outcome(pedis, outcome = "any_amputation")
sheets_write(data = desc_any,
ss = googlesheet_id,
sheet = "descr-table-any")
desc_maj <- descriptive_table_outcome(pedis, outcome = "major_amputation")
sheets_write(data = desc_maj,
ss = googlesheet_id,
sheet = "descr-table-maj")
# Setting model formula
f_any <- as.formula("any_amputation ~ p + e_ordinal_5 + d + i + s + alter_bei_aufnahme + gender")
f_major <- as.formula("major_amputation ~ p + e_ordinal_5 + d + i + s + alter_bei_aufnahme + gender")
# Setting priors
# Uninformed priors
uninformed_prior <- cauchy(location = c(rep(0, 5), 0, 0), scale = c(rep(1, 5), .5, .5))
source("./scripts/compute-informed-prior.R")
# Informed priors (from Pickwell et al.)
tibble(pooled_beta, pooled_sd) %>%
pivot_longer(cols = c("pooled_beta", "pooled_sd")) %>%
mutate(exp = exp(value)) %>%
set_names("statistic", "non-exponentiated", "exponentiated")
## # A tibble: 2 x 3
## statistic `non-exponentiated` exponentiated
## <chr> <dbl> <dbl>
## 1 pooled_beta 0.535 1.71
## 2 pooled_sd 0.273 1.31
informed_prior <- cauchy(location = c(rep(pooled_beta, 5), 0, 0), scale = c(rep(pooled_sd, 5), .5, .5))
# Set number of MCMC interations
n_iter <- 4e4
seed <- 3412
# Outcome: Any Amputation
# Approach: Non-Informed
noninformed_any <- fit_model(n_iter = n_iter, exponentiate = FALSE, seed = seed)
# Posterior Histogram with prior
errorbar_1 <- plot(noninformed_any[[1]]) + ggtitle("Non Informed Any")
# AUC Posterior
auc_noninformed_any <- posterior_auc(model = noninformed_any,
data = pedis,
outcome = "any_amputation")
# Outcome: Any Amputation
# Approach: Informed Prior
informed_any <- fit_model(formula = f_any, prior = informed_prior, n_iter = n_iter, seed = seed)
# Posterior histograms with priors
errorbar_2 <- plot(informed_any[[1]]) + ggtitle("Informed Any")
# AUC Posterior
auc_informed_any <- posterior_auc(model = informed_any,
data = pedis,
outcome = "any_amputation")
# Outcome: Any Amputation
# Approach: Informed Prior
noninformed_major <- fit_model(formula = f_major, n_iter = n_iter, seed = seed)
# Posterior Histogram with priors
errorbar_3 <- plot(noninformed_major[[1]]) + ggtitle("Non-Informed Major")
# AUC Posterior
auc_noninformed_major <- posterior_auc(model = noninformed_major,
data = pedis,
outcome = "major_amputation")
# Outcome: Major Amputation
# Approach: Informed Prior
informed_major <- fit_model(formula = f_major,
prior = informed_prior,
seed = seed,
n_iter = n_iter)
# Posterior histograms with priors
errorbar_4 <- plot(informed_major[[1]]) + ggtitle("Informed Major")
# AUC Posterior
auc_informed_major <- posterior_auc(model = informed_major,
data = pedis,
outcome = "major_amputation")
auc <- list("Any Amputation - Non-Informed" = auc_noninformed_any,
"Any Amputation - Informed" = auc_informed_any,
"Major Amputation - Non-Informed" = auc_noninformed_major,
"Major Amputation - Informed" = auc_informed_any)
# Descriptive Summary
auc_summarised <- map(auc, auc_summary) %>%
bind_rows(.id = "Model") %>%
mutate(HDI = paste0("[", myround(lower, 3), "-", myround(upper, 3), "]")) %>%
mutate_if(is.numeric, myround, 3)
auc_summarised %>%
select(Model, med, HDI) %>%
kable(digits = 3)
Model | med | HDI |
---|---|---|
Any Amputation - Non-Informed | 0.793 | [0.778-0.801] |
Any Amputation - Informed | 0.790 | [0.774-0.802] |
Major Amputation - Non-Informed | 0.765 | [0.725-0.779] |
Major Amputation - Informed | 0.790 | [0.774-0.802] |
sheets_write(data = auc_summarised,
ss = googlesheet_id,
sheet = "auc")
# Plot Posterior AUC Distributions
posterior_auc_distributions <- auc %>% map(hist, bw = .005)
get_y_axis <- function(plt_obj) ggplot_build(plt_obj)$layout$panel_scales_y[[1]]$range$range
y_max <- map_df(posterior_auc_distributions, get_y_axis) %>%
pivot_longer(cols = names(.)) %>%
pull(value) %>%
max %>%
`+`(2)
x_min <- min(map_dbl(auc, quantile, .02))
x_max <- max(map_dbl(auc, max)) + .01
x_axis <- scale_x_continuous(name = "AUC value",
limits = c(x_min, x_max),
breaks = seq(0, 1, by = .02))
y_axis <- scale_y_continuous(name = "Density", limits = c(0, y_max))
g <- ggpubr::ggarrange(
posterior_auc_distributions[[1]] +
x_axis +
y_axis +
rremove("x.title"),
posterior_auc_distributions[[2]] +
x_axis +
y_axis +
rremove("x.title") +
rremove("y.title"),
posterior_auc_distributions[[3]] +
x_axis +
y_axis,
posterior_auc_distributions[[4]] +
x_axis +
y_axis +
rremove("y.title"),
ncol = 2,
nrow = 2,
labels = c("A", "B", "C", "D"))
g
ggsave(plot = g,
filename = "img/posterior-auc-values.png",
unit = "cm", width = 30, height = 30,
dpi = 320,
scale = .5)
posterior_errorbars <- gridExtra::grid.arrange(
errorbar_1 + scale_y_continuous(breaks = 0:10) + ggtitle("Any Amputation - Non-Informed."),
errorbar_2 + scale_y_continuous(breaks = 0:10) + ggtitle("Any Amputation - Informed"),
errorbar_3 + scale_y_continuous(breaks = 0:10) + ggtitle("Major Amputation - Non-Informed"),
errorbar_4 + scale_y_continuous(breaks = 0:10) + ggtitle("Major Amputation - Informed"))
ggsave(posterior_errorbars,
filename = "img/posterior-parameter-errorbars.png",
unit = "cm",
width = 20,
height = 20)
models <- list("Any Amputation - Non-Informed" = noninformed_any,
"Any Amputation - Informed" = informed_any,
"Major Amputation - Non-Informed" = noninformed_major,
"Major Amputation - Informed" = informed_major)
summary_models <- map(models, summary_coefs) %>% bind_rows(.id = "model")
sheets_write(data = summary_models,
ss = googlesheet_id,
sheet = "coefficients")
xlimit <- c(-1.2, 3)
ylimit <- c(0, 2.4)
hist_posterior <- models %>%
map(pluck, 3) %>%
map(function(x) x +
# scale_x_continuous("Value of Coefficients", limits = c(-1.2, 3)) +
theme(strip.text = element_text(size = 8),
axis.text = element_text(size = 8)) +
rremove("x.title") +
rremove("y.title") +
coord_cartesian(ylim = ylimit)
)
hist_plot <- ggarrange(hist_posterior[[1]],
hist_posterior[[2]],
hist_posterior[[3]],
hist_posterior[[4]],
labels = LETTERS[1:4],
ncol = 4)
hist_plot
ggsave(plot = hist_plot,
filename = "img/posterior-beta-values.png",
unit = "cm", width = 22, height = 18,
dpi = 300)
filename <- names(models) %>%
tolower %>%
gsub(pattern = "\\s+", replacement = " ") %>%
gsub(pattern = "\\s{1}", replacement = "-") %>%
gsub(pattern = "---", replacement = "-") %>%
paste0("models/", ., ".RDS")
map2(models, filename, saveRDS)
Exemplary predictions of 6 month amputation incidence
mat_mcmc <- noninformed_any[[2]] %>% as.matrix()
x1 <- mat_mcmc %*% c(intercept = 1, p = 2, e = 2, d = 1, i = 1, s = 1, age = 60, gender = 1) %>%
c %>%
enframe(name = NULL)
x2 <- mat_mcmc %*% c(intercept = 1, p = 3, e = 3, d = 3, i = 1, s = 1, age = 60, gender = 1) %>%
c %>%
enframe(name = NULL)
data <- bind_rows(list("Patient with low PEDIS classification" = x1,
"Patient with high PEDIS classification" = x2),
.id = "patient") %>%
mutate(value = 1 / (1 + exp(-1 * value))) %>%
mutate(patient = factor(x = patient,
levels = c("Patient with low PEDIS classification",
"Patient with high PEDIS classification"),
labels = c("PEDIS classification: P2, E2, D1, I1, S1",
"PEDIS classification: P3, E3, D3, I1, S1"),
ordered = TRUE))
mean_summary <- data %>%
group_by(patient) %>%
summarise(avg = mean(value), med = median(value))
predictions_summary <- data %>%
group_by(patient) %>%
nest() %>%
mutate(data = map(data, function(df) pull(df, 1))) %>%
mutate(hdi = map(data, function(x) tidy_hdi(x)),
med = map_dbl(data, function(x) median(x)),
avg = map_dbl(data, function(x) mean(x))) %>%
unnest(cols = c(hdi)) %>%
select(-data)
predictions_summary %>%
ggplot(aes(xmin = lower, xmax = upper, x = med, y = 1, group = patient)) +
geom_histogram(aes(x = value,
y = ..density..,
fill = patient),
binwidth = .01,
data = data,
inherit.aes = FALSE) +
geom_errorbarh(size = 1.5) +
geom_point(shape = 20, size = 6) +
scale_y_continuous("Density") +
scale_x_continuous("Predicted risk (six-month amputation incidence)",
breaks = seq(0, 1, by = .2),
limits = c(-.1, 1.05)) +
ggtitle(label = "Posterior Distribution of Predictions",
subtitle = "Predictions based on any-amputation risk model") +
scale_fill_manual(values = c("grey50", "grey70")) +
scale_color_manual(values = c("black", "black"), guide = F) +
theme_minimal() +
theme(title = element_text(size = 14),
legend.position = "bottom",
legend.direction = "vertical",
legend.title = element_blank(),
text = element_text(size = 18))
ggsave(filename = "./img/post-amp-any-prob.png", units = "cm", width = 20, height = 20)
mat_mcmc <- noninformed_major[[2]] %>% as.matrix()
x1 <- mat_mcmc %*% c(intercept = 1, p = 2, e = 2, d = 1, i = 1, s = 1, age = 60, gender = 1) %>%
c %>%
enframe(name = NULL)
x2 <- mat_mcmc %*% c(intercept = 1, p = 3, e = 3, d = 3, i = 1, s = 1, age = 60, gender = 1) %>%
c %>%
enframe(name = NULL)
data <- bind_rows(list("Patient with low PEDIS Score" = x1,
"Patient with high PEDIS Score" = x2),
.id = "patient") %>%
mutate(value = 1 / (1 + exp(-1 * value))) %>%
mutate(patient = factor(x = patient,
levels = c("Patient with low PEDIS Score", "Patient with high PEDIS Score"),
ordered = TRUE))
mean_summary <- data %>%
group_by(patient) %>%
summarise(avg = mean(value))
data %>%
ggplot(aes(x = value, y = ..density.., fill = patient)) +
geom_histogram(binwidth = .01) +
scale_y_continuous("Density") +
scale_x_continuous("Probability", limits = c(-0.05, .5)) +
geom_vline(data = mean_summary,
aes(xintercept = avg, col = patient),
linetype = 5,
size = 1.5) +
ggtitle("Histogram of Posterior Amputation Probability") +
scale_fill_manual(values = c("grey30", "grey50")) +
scale_color_manual(values = c("black", "black"), guide = F) +
theme_minimal() +
theme(legend.position = "bottom",
legend.title = element_blank())
ggsave(filename = "./img/post-amp-major-prob.png", units = "cm", width = 22, height = 22)
plot_mcmc <- function(model, color_scheme = "gray", outcome = "any_amputation") {
posterior <- as.array(model)
color_scheme_set(color_scheme)
bayesplot::mcmc_trace(posterior, pars = c("p", "e_ordinal_5", "d", "i", "s"),
facet_args = list(ncol = 1, strip.position = "top")) +
theme_pubr() +
scale_y_continuous(limits = c(-0.6, 2)) +
theme(legend.position = "left")
ggsave(filename = paste0("img/mcmc-trace-posterior-", outcome, ".png"))
}
models <- list(noninformed_any, informed_any, noninformed_major, informed_major)
models_title <- c("amp-any-noninfo", "amp-any-info", "amp-maj-noninfo", "amp-major-info")
map2(models, models_title, function(m, col) plot_mcmc(pluck(m, 4), outcome = col))
hypos <- brms::hypothesis(noninformed_any[[4]], "d > 0")
sum_any <- summary(noninformed_any[[4]], digits = 3)
auc_df <- auc %>%
enframe %>%
unnest(value)
cohen <- auc_df %>%
filter(grepl(pattern = "Non", x = name)) %>%
effsize::cohen.d(value ~ name, data = .) %>%
pluck("estimate")
delta <- auc_df %>%
filter(grepl(pattern = "Non", x = name)) %>%
group_by(name) %>%
summarise(avg = mean(value)) %>%
pull(avg) %>%
diff
enframe(c(cohen = cohen, delta = delta)) %>%
kable(digits = 3) %>%
kableExtra::kable_styling(bootstrap_options = "striped", full_width = F)
name | value |
---|---|
cohen | 2.417 |
delta | -0.031 |
cohen <- auc_df %>%
filter(grepl(pattern = "Major", x = name)) %>%
effsize::cohen.d(value ~ name, data = .) %>%
pluck("estimate")
delta <- auc_df %>%
filter(grepl(pattern = "Major", x = name)) %>%
group_by(name) %>%
summarise(avg = mean(value)) %>%
pull(avg) %>%
diff
enframe(c(cohen = cohen, delta = delta)) %>%
kable(digits = 3) %>%
kableExtra::kable_styling(bootstrap_options = "striped", full_width = F)
name | value |
---|---|
cohen | 2.217 |
delta | -0.029 |
pedis_class <- pedis %>% select(p, e_ordinal_5, d, i, s)
cor(pedis_class, method = "kendall")
## p e_ordinal_5 d i s
## p 1.000000000 0.2993098 0.25199153 0.004545883 0.07785416
## e_ordinal_5 0.299309783 1.0000000 0.33726716 0.232923072 -0.06776940
## d 0.251991534 0.3372672 1.00000000 0.364399661 0.09786555
## i 0.004545883 0.2329231 0.36439966 1.000000000 0.04830705
## s 0.077854164 -0.0677694 0.09786555 0.048307049 1.00000000
fit <- rstanarm::stan_glm(any_amputation ~ i, family = "binomial", data = pedis)
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fit <- rstanarm::stan_glm(major_amputation ~ i, family = "binomial", data = pedis)
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summary(fit)
##
## Model Info:
##
## function: stan_glm
## family: binomial [logit]
## formula: major_amputation ~ i
## algorithm: sampling
## priors: see help('prior_summary')
## sample: 4000 (posterior sample size)
## observations: 237
## predictors: 2
##
## Estimates:
## mean sd 2.5% 25% 50% 75% 97.5%
## (Intercept) -2.7 0.5 -3.8 -3.1 -2.7 -2.3 -1.7
## i 0.3 0.2 -0.1 0.2 0.3 0.5 0.8
## mean_PPD 0.1 0.0 0.1 0.1 0.1 0.1 0.2
## log-posterior -92.1 1.0 -94.9 -92.5 -91.8 -91.4 -91.1
##
## Diagnostics:
## mcse Rhat n_eff
## (Intercept) 0.0 1.0 2159
## i 0.0 1.0 2348
## mean_PPD 0.0 1.0 2789
## log-posterior 0.0 1.0 1306
##
## For each parameter, mcse is Monte Carlo standard error, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence Rhat=1).
outcome <- c("any_amputation", "major_amputation")
predictors <- c("p", "e_ordinal_5", "d", "i", "s", "alter_bei_aufnahme", "gender")
formulas <- expand.grid(outcome = outcome, predictors = predictors) %>%
as_tibble() %>%
unite(col = formula, c("outcome", "predictors"), sep = " ~ ") %>%
pull(1) %>%
as.list()
univar_association <- function(formula) {
set.seed(1231238)
model <- rstanarm::stan_glm(formula = as.formula(formula),
family = "binomial",
data = pedis,
prior = cauchy(location = 0, scale = 0.5))
intervals <- hdi(x = model, ci = .95) %>%
as.tibble() %>%
filter(!grepl(pattern = "Intercept", x = Parameter))
params <- as.matrix(x = model) %>%
as.tibble() %>%
select(2) %>%
summarise_all(list(odds_ratio = function(x) exp(median(x)),
coefficient = function(y) median(y)))
posterior <- as.matrix(model) %>%
as.tibble %>%
rename(values = 2) %>%
select(2) %>%
nest(data = everything())
outcome <- str_extract(string = formula, pattern = "(.*)_amputation")
outcome <- enframe(outcome, name = "outcome")
bind_cols(params, intervals, posterior, outcome)
}
univariate_coefficients <- map_df(formulas, univar_association)
##
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## SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 4).
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## SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 4).
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## SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 1).
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lvls <- c("p", "e_ordinal_5", "d", "i", "s", "alter_bei_aufnahme", "gender")
lbls <- c("Perfusion",
"Extent",
"Depth",
"Infection",
"Sensation",
"Age",
"Gender")
univar_analysis <- univariate_coefficients %>%
arrange(value) %>%
mutate(Parameter = factor(x = Parameter,
levels = lvls,
labels = lbls,
ordered = TRUE)) %>%
mutate(value = factor(x = value,
levels = c("any_amputation", "major_amputation"),
labels = c("Any Amputation", "Major Amputation"),
ordered = TRUE)) %>%
mutate(CI_low_exp = exp(CI_low), CI_high_exp = exp(CI_high))
univar_analysis %>%
ggplot(aes(x = forcats::fct_rev(Parameter), y = coefficient, ymin = CI_low, ymax = CI_high)) +
geom_point() +
geom_errorbar(width = .2) +
geom_abline(intercept = 0, slope = 0, size = .2, linetype = 2) +
facet_wrap(~ value, ncol = 1) +
coord_flip(clip = "off") +
scale_y_continuous("Coefficients", breaks = seq(-1, 3, by = .5)) +
theme_bw() +
theme(axis.title.y = element_blank()) +
ggtitle("Coefficients of Univariate Logistic Models")
# Din A4 210 × 297 millimeters (Half)
ggsave(filename = "img/appendix-univar-models.png",
units = "cm",
height = 29.7 / 2, width = 21.0 / 2,
dpi = 300)
univar_analysis <- univar_analysis %>%
mutate_if(is.numeric, sprintf, fmt = "%1.3f") %>%
mutate(coef_hdi = glue::glue("{coefficient} [{CI_low} to {CI_high}]")) %>%
mutate(or_hdi = glue::glue("{odds_ratio} [{CI_low_exp} to {CI_high_exp}]")) %>%
mutate_at("coef_hdi", as.character) %>%
select(Parameter, value, coef_hdi, or_hdi, starts_with("CI"), odds_ratio, coefficient)
sheets_write(data = univar_analysis %>% arrange(value),
ss = googlesheet_id,
sheet = "univar-analysis")
We created Bayesian Logistic Regression models using the open-source rstanarm package for the statistical programming language R.
Our model analyzes the relationship between the PEDIS-system, a clinical classification to describe characteristics of Diabetic Foot Ulcers, and amputation, which results from such ulcers as a severe complication.
We used two distinct amputation definitions:
Furthermore, we scrutinized the impact of an informed prior compared to a non-informed one.
As a result, we created four distinct model.
model <- readRDS("models/any-amputation-non-informed.RDS") %>% pluck(4)
color_scheme_set("viridis")
# Effect size
summary(model) %>%
as.data.frame() %>%
as_tibble(rownames = "Predictors") %>%
select(Predictors, n_eff) %>%
slice(1:8) %>%
kable() %>%
kableExtra::kable_styling(bootstrap_options = "striped", full_width = F)
Predictors | n_eff |
---|---|
(Intercept) | 26516 |
p | 24728 |
e_ordinal_5 | 25087 |
d | 26378 |
i | 28326 |
s | 32397 |
alter_bei_aufnahme | 28275 |
gender | 31458 |
# MCMC trace
mcmc_trace(model)
# Scatterplots of MCMC draws
mcmc_pairs(x = model, pars = c("p", "e_ordinal_5", "d", "i", "s"))
# Posterior predictive check
pp_check(model)
model <- readRDS("models/major-amputation-non-informed.RDS") %>% pluck(4)
color_scheme_set("viridis")
# Effect size
summary(model) %>%
as.data.frame() %>%
as_tibble(rownames = "Predictors") %>%
select(Predictors, n_eff) %>%
slice(1:8) %>%
kable() %>%
kableExtra::kable_styling(bootstrap_options = "striped", full_width = F)
Predictors | n_eff |
---|---|
(Intercept) | 21569 |
p | 26686 |
e_ordinal_5 | 22283 |
d | 23718 |
i | 28822 |
s | 30906 |
alter_bei_aufnahme | 29617 |
gender | 30830 |
# MCMC trace
mcmc_trace(model)
# Scatterplots of MCMC draws
mcmc_pairs(x = model, pars = c("p", "e_ordinal_5", "d", "i", "s"))
# Posterior predictive check
pp_check(model)
model <- readRDS("models/any-amputation-informed.RDS") %>% pluck(4)
color_scheme_set("viridis")
# Effect size
summary(model) %>%
as.data.frame() %>%
as_tibble(rownames = "Predictors") %>%
select(Predictors, n_eff) %>%
slice(1:8) %>%
kable() %>%
kableExtra::kable_styling(bootstrap_options = "striped", full_width = F)
Predictors | n_eff |
---|---|
(Intercept) | 25945 |
p | 29933 |
e_ordinal_5 | 23116 |
d | 27226 |
i | 26024 |
s | 33476 |
alter_bei_aufnahme | 28278 |
gender | 30840 |
# MCMC trace
mcmc_trace(model)
# Scatterplots of MCMC draws
mcmc_pairs(x = model, pars = c("p", "e_ordinal_5", "d", "i", "s"))
# Posterior predictive check
pp_check(model)
model <- readRDS("models/major-amputation-informed.RDS") %>% pluck(4)
color_scheme_set("viridis")
# Effect size
summary(model) %>%
as.data.frame() %>%
as_tibble(rownames = "Predictors") %>%
select(Predictors, n_eff) %>%
slice(1:8) %>%
kable() %>%
kableExtra::kable_styling(bootstrap_options = "striped", full_width = F)
Predictors | n_eff |
---|---|
(Intercept) | 18853 |
p | 28542 |
e_ordinal_5 | 17698 |
d | 24427 |
i | 28594 |
s | 31812 |
alter_bei_aufnahme | 29640 |
gender | 31760 |
# MCMC trace
mcmc_trace(model)
# Scatterplots of MCMC draws
mcmc_pairs(x = model, pars = c("p", "e_ordinal_5", "d", "i", "s"))
# Posterior predictive check
pp_check(model)